Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.8% → 100.0%

Time: 8.1s

Alternatives: 8

Speedup: 1.3×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-4, \frac{y - x}{z}, -2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma -4.0 (/ (- y x) z) -2.0))

C

double code(double x, double y, double z) {
	return fma(-4.0, ((y - x) / z), -2.0);
}

Julia

function code(x, y, z)
	return fma(-4.0, Float64(Float64(y - x) / z), -2.0)
end

Wolfram

code[x_, y_, z_] := N[(-4.0 * N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision] + -2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y - x}{z}, -2\right)} \]
6. Add Preprocessing

Alternative 2: 65.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ t_2 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z))
        (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z))
        (t_2 (/ (* -4.0 y) z)))
   (if (<= t_1 -5e+252)
     t_0
     (if (<= t_1 -5e+93)
       t_2
       (if (<= t_1 -2e+22)
         t_0
         (if (<= t_1 -1.0) -2.0 (if (<= t_1 2e+124) t_0 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = t_0;
	} else if (t_1 <= -5e+93) {
		tmp = t_2;
	} else if (t_1 <= -2e+22) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+124) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * 4.0d0) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    t_2 = ((-4.0d0) * y) / z
    if (t_1 <= (-5d+252)) then
        tmp = t_0
    else if (t_1 <= (-5d+93)) then
        tmp = t_2
    else if (t_1 <= (-2d+22)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+124) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -5e+252) {
		tmp = t_0;
	} else if (t_1 <= -5e+93) {
		tmp = t_2;
	} else if (t_1 <= -2e+22) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+124) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	t_2 = (-4.0 * y) / z
	tmp = 0
	if t_1 <= -5e+252:
		tmp = t_0
	elif t_1 <= -5e+93:
		tmp = t_2
	elif t_1 <= -2e+22:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+124:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	t_2 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (t_1 <= -5e+252)
		tmp = t_0;
	elseif (t_1 <= -5e+93)
		tmp = t_2;
	elseif (t_1 <= -2e+22)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+124)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	t_2 = (-4.0 * y) / z;
	tmp = 0.0;
	if (t_1 <= -5e+252)
		tmp = t_0;
	elseif (t_1 <= -5e+93)
		tmp = t_2;
	elseif (t_1 <= -2e+22)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+124)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+252], t$95$0, If[LessEqual[t$95$1, -5e+93], t$95$2, If[LessEqual[t$95$1, -2e+22], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+124], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4.9999999999999997e252 or -5.0000000000000001e93 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2e22 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.9999999999999999e124

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{4 \cdot \frac{x}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      3. *-lowering-*.f64 61.5

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

   if -4.9999999999999997e252 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5.0000000000000001e93 or 1.9999999999999999e124 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. *-lowering-*.f64 61.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

   if -2e22 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 91.6%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+252}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - y\right)}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x y)) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -2e+22)
     t_0
     (if (<= t_1 5000000.0) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - y)) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -2e+22) {
		tmp = t_0;
	} else if (t_1 <= 5000000.0) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - y)) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -2e+22)
		tmp = t_0;
	elseif (t_1 <= 5000000.0)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+22], t$95$0, If[LessEqual[t$95$1, 5000000.0], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2e22 or 5e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{4 \cdot \color{blue}{\left(x - y\right)}}{z} \]
   4. Step-by-step derivation

      1. --lowering--.f64 100.0

         \[\leadsto \frac{4 \cdot \color{blue}{\left(x - y\right)}}{z} \]

   5. Simplified 100.0%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(x - y\right)}}{z} \]

   if -2e22 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5e6

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y - x}{z}, -2\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2} \]

      3. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2 \cdot 1} \]

      4. lft-mult-inverse N/A

         \[\leadsto -4 \cdot \frac{y}{z} + -2 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      5. associate-*l* N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(-2 \cdot \frac{1}{x}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{x}\right) \cdot x \]

      7. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)} \cdot x \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \left(\color{blue}{-2} \cdot \frac{1}{x}\right) \cdot x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2 \cdot \left(\frac{1}{x} \cdot x\right)}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, -2 \cdot \color{blue}{1}\right) \]

      14. metadata-eval 95.8

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2}\right) \]

   8. Simplified 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 y) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -10.0) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) * y) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-10.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 * y) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -10.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * y) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 * y) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -10 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. *-lowering-*.f64 51.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

   if -10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 95.8%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ -4.0 z))) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -10.0) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * ((-4.0d0) / z)
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-10.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-4.0 / z)
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -10.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-4.0 / z))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-4.0 / z);
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(-4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -10 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. *-lowering-*.f64 51.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{-4 \cdot y}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(-4 \cdot y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot -4\right) \cdot y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot -4\right) \cdot y} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -4}{z}} \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-4}}{z} \cdot y \]

      7. /-lowering-/.f64 50.9

         \[\leadsto \color{blue}{\frac{-4}{z}} \cdot y \]

   7. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\frac{-4}{z} \cdot y} \]

   if -10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 95.8%

      \[\leadsto \color{blue}{-2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -10:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 4.0 (/ x z) -2.0)))
   (if (<= x -1.8e+54) t_0 (if (<= x 1.15e+39) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(4.0, (x / z), -2.0);
	double tmp;
	if (x <= -1.8e+54) {
		tmp = t_0;
	} else if (x <= 1.15e+39) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(4.0, Float64(x / z), -2.0)
	tmp = 0.0
	if (x <= -1.8e+54)
		tmp = t_0;
	elseif (x <= 1.15e+39)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[x, -1.8e+54], t$95$0, If[LessEqual[x, 1.15e+39], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8000000000000001e54 or 1.15000000000000006e39 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{-2} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

      10. /-lowering-/.f64 87.1

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z}}, -2\right) \]

   5. Simplified 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

   if -1.8000000000000001e54 < x < 1.15000000000000006e39

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y - x}{z}, -2\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2} \]

      3. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2 \cdot 1} \]

      4. lft-mult-inverse N/A

         \[\leadsto -4 \cdot \frac{y}{z} + -2 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      5. associate-*l* N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(-2 \cdot \frac{1}{x}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{x}\right) \cdot x \]

      7. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)} \cdot x \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \left(\color{blue}{-2} \cdot \frac{1}{x}\right) \cdot x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2 \cdot \left(\frac{1}{x} \cdot x\right)}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, -2 \cdot \color{blue}{1}\right) \]

      14. metadata-eval 90.6

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2}\right) \]

   8. Simplified 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -1.5e+134) t_0 (if (<= x 2.2e+94) (fma -4.0 (/ y z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double tmp;
	if (x <= -1.5e+134) {
		tmp = t_0;
	} else if (x <= 2.2e+94) {
		tmp = fma(-4.0, (y / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	tmp = 0.0
	if (x <= -1.5e+134)
		tmp = t_0;
	elseif (x <= 2.2e+94)
		tmp = fma(-4.0, Float64(y / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -1.5e+134], t$95$0, If[LessEqual[x, 2.2e+94], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999998e134 or 2.20000000000000012e94 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{4 \cdot \frac{x}{z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      3. *-lowering-*.f64 78.0

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

   if -1.49999999999999998e134 < x < 2.20000000000000012e94

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y - x}{z}, -2\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2} \]

      3. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2 \cdot 1} \]

      4. lft-mult-inverse N/A

         \[\leadsto -4 \cdot \frac{y}{z} + -2 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      5. associate-*l* N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(-2 \cdot \frac{1}{x}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{x}\right) \cdot x \]

      7. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)} \cdot x \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-4, \color{blue}{\frac{y}{z}}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \left(\color{blue}{-2} \cdot \frac{1}{x}\right) \cdot x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2 \cdot \left(\frac{1}{x} \cdot x\right)}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, -2 \cdot \color{blue}{1}\right) \]

      14. metadata-eval 87.5

          \[\leadsto \mathsf{fma}\left(-4, \frac{y}{z}, \color{blue}{-2}\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Simplified 30.0%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))